well... kind of. the (spoilers!) dragonets of destiny's parents (a few of them) sold or just let the talons of peace have them. and quibli's mother neglected and eventually gave up her son
I mean, since the dragons are technically animals and all of the dragonets have been (spoiler alert!) abandoned in some way, then maybe this is a yes. (Though Sunny may not have been abandoned, as it describes in book 5.)
in Dragonslayer (not exactly a spinoff, just the story from the humans/scavengers perspective) one of the main characters Wren kills a rabbit with a slingshot, but her dragon friend does not eat it and he cries when given the corpse. and in the winglets collection book (specifically #3, deserter) there's a brief mention of Princess Burn cutting the legs off of jackrabbits to see what they do. neither scenes necessarily go into detail, but they happen
No cats are shown or mentioned to have died in Wings of Fire, but there are implications of it being a common occurrence; in book 10, a dragonet owns a pet cat and Quibli thinks that the dragonet must be a person of some power since usually the cat would have been eaten by then.
It is also said, I can’t quite remember in which book(perhaps book 5?), that pets are usually accidentally eaten by family members of the dragon attempting to keep them.
In Legends Dragonslayer, I believe(but I could be misremembering) that Wren comments on how a cat named Dragon was likely to be eaten when faced by an actual dragon. In short, no, no cats die in Wings of Fire but it is hinted at and could still be triggering.
There is most likely more mentions and appearances of snakes, but in the fifth book, two snakes appear, one of them biting and killing one of the characters there.
there's a few mentions of sharks throughout the books, but they've never been presented in a scary light and have more been classified as food than a threat
One of the minor antagonists, Whirlpool, attempts to groom a child princess in hopes of becoming king. However, he fails and is killed by his victim. Additionally, Glory dates a dragon who may be double her age, although it's unclear because different books have somewhat contradictory information.
No, but a few mentions of 'breeding programs'; a dragoness at one point comments that she was ordered to breed in order to produce soldiers for her queen. Also, in the second book, a male dragon mistakenly believes one of the main characters is showing romantic interest in him due to communication issues and tries to act on it, but nothing further happens.
a dragon named Jerboa III is missing claws on her front talons because of a curse, so technically. in the third arc, there's also mentions of initials being carved into talon palms for organizing purposes
Several instances, for example a scavenger (human) getting his head bitten off by a dragon, who comments on its flavor and texture, in Book 1 — As well as the decapitated head of a MudWing, which was enchanted to look like the character Glory’s head, being thrown at a crowd of dragons in Book 8. I’m sure there’s several other instances which I can’t think of right now.
There is one character that is missing a limb due to being in the war, though it is heavily implied that this is a war injury as opposed to purposeful amputation.
(MILDLY GRAPHIC COMMENT) In one book (Darkstalker: Legends), Darkstalker uses magic to have his father kill himself by cutting out his tongue and spilling his internal organs onto the ground.
And in the first book, Queen Scarlet gets sprayed by corrosive spit on the side of her face, which does leave a scar.
…
ALSO in another book, a dragon accidently gets his eye scratched by a venomous tail.
Sometimes there are things characters think are ghosts but as of now, there have been no spirits of any kind that have appeared in the wings of fire books
The main plot of the Lost Continent arc (books 11-15) involves people being mind-controlled using a plant, which allows multiple antagonists to possess their bodies against their will. The victims of this range from infants to adults and some animals.
While there are no medical syringes or needles, in the Lost Continent arc (books 11-15), multiple people are stabbed and injected with various toxins or venoms using the Hivewing's stingers. These seem to be a cross between wasp stingers and needles.
while not usually for depressive reasons theres multiple times of self inflicted injuries in the series
Spoilers ahead -
book 7 a character burns themself with lava to stay awake -
book 11 a character mentions having hurt themself in an attempt to break out of possession -
book 13 a character claws themself while possessed -
Darkstalker legends a character cuts themself for a blood oath and immediately has thoughts of doing more
its not explicitly labeled, but in Darkstalker (prequel book), his brother Whiteout is described making a keening/wailing sound in response to distressing information, and is later said to be sobbing, which i believe could be classified as a meltdown
One of the protagonists, Moon, can sometimes feel overwhelmed by her powers and it can lead to her blacking out slightly as her powers overcome her, but it's not really an anxiety attack and it's to show how she's never been taught anything about her powers.
in book fifteen there is a poorly handled assisted suicide. also while not actual suicides still may be triggering - in book one there’s an offhand comment about how it was surprising a disabled character hasn’t killed themself, in Darkstalker a character is magically forced to kill themself
Some LGBT characters die, but there are many other happy LGBT characters in the series who live and even a main focus lesbian couple with one of them being a POV character in the hidden kingdom series
Starflight has a crush on one of the other members of the Dragonets of Destiny, who were kidnapped together and raised by the same "guardians" from birth. Sunny rejects his advances because she sees him as her brother.